On Normal $\mu$-Hankel Operators

Abstract

Hankel operators have numerous realizations and form one of the most important classes of operators in the spaces of analytic functions. These operators can be defined as those having Hankel matrices (i.e. matrices whose entries depend only on the sum of the indices) with respect to some orthonormal basis for a separable Hilbert space. This paper continues the authors’ research of 2021 which introduced a new class of operators in Hilbert spaces; i.e., $\mu$-Hankel operators, with $\mu$ a complex parameter. These operators act in a separable Hilbert space and have matrices in some orthonormal basis of the space whose diagonals orthogonal to the main diagonal present geometric progressions with common ratio $\mu$. Thus, the classical Hankel operators correspond to the case $\mu=1$. The main result of the article is the normality criterion for $\mu$-Hankel operators. By analogy with Hankel operators, the class of operators under consideration has concrete realizations in the form of integral operators which enables us to apply the abstract results, and thereby contribute to the theory of integral operators. We consider an example of these realizations in the Hardy space on the unit disk. Also, we give some criteria for the self-adjointness and normality of these operators.